1 Modal Logic. 1.1 Basics

Size: px
Start display at page:

Download "1 Modal Logic. 1.1 Basics"

Transcription

1 1 Modal Logic As mentioned in the preface, we assume familiarity with the basic definitions concerning the syntax and semantics of modal logic. The purpose of this first chapter is to briefly recall notation and terminology. We focus on some aspects of modal logic that feature prominently in its extensions with fixpoint operators. Convention 1.1 Throughout this text we let Prop be a countably infinite set of propositional variables, whose elements are usually denoted as p, q, r, x, y, z,..., and we let D be a finite set of (atomic) actions, whose elements are usually denoted as d, e, c,.... We will usually focus on a finite subset P of Prop, consisting of those propositional variables that occur freely in a particular formula. In practice we will often suppress explicit reference to Prop, P and D. 1.1 Basics Structures I Introduce LTSs as process graphs Definition 1.2 A (labelled) transition system, LTS, or Kripke model of type (P, D) is a triple S = hs, V, Ri such that S is a set of objects called states or points, V : P! }(S)isavaluation, and R = {R d S S d 2 D} is a family of binary accessibility relations. Elements of the set R d [s] :={t 2 S (s, t) 2 R d } are called d-successors of s. A transition system is called image-finite or finitely branching if R d [s] is finite, for every d 2 D and s 2 S. A pointed transition system or Kripke model is a pair (S,s) consisting of a transition system S and a designated state s in S. Remark 1.3 It will be convenient to work with an alternative, coalgebraic presentation of transition systems. Intuitively, it should be clear that instead of having a valuation V : P! }(S), telling us at which states each proposition letter is true, we could just as well have a marking V : S! }(P) informing us which proposition letters are true at each state. Also, a binary relation R on a set S can be represented as a map R[ ] :S! }(S) mapping a state s to the collection R[s] of its successors. In this line, a family R = {R d S S d 2 D} of accessibility relations can be seen as a map R : S! }(S) D,where}(S) D denotes the set of maps from D to }(S). Combining these two maps into one single function, we see that a transition system S = hs, V, Ri of type (P, D) can be seen as a pair hs, i, where : S! }(P) }(S) D is the map given by (s) :=( V (s), R(s)). For future reference we define the notion of a Kripke functor. Definition 1.4 Fix a set P of proposition letters and a set D of atomic actions. Given a set S, letk D,P S denote the set K D,P S := }(P) }(S) D. This operation will be called the Kripke functor associated with D and P.

2 4 Modal logic A typical element of K D,P S will be denoted as (, X), with P and X = {X d d 2 D} with X d S for each d 2 D. When we take this perspective we will sometimes refer to Kripke models as K D,P S- coalgebras or Kripke coalgebras. Given this definition we may summarize Remark 1.3 by saying that any transition system can be presented as a pair S = hs, : S! KSi where K is the Kripke functor associated with S. In practice, we will usually write K rather than K D,P. Syntax Working with fixpoint operators, we may benefit from a set-up in which the use of the negation symbol may only be applied to atomic formulas. The price that one has to pay for this is an enlarged arsenal of primitive symbols. In the context of modal logic we then arrive at the following definition. Definition 1.5 The language ML D of polymodal logic in D is defined as follows: ' ::= p p? > ' _ ' ' ^ ' 3 d ' 2 d ' where p 2 Prop, and d 2 D. Elements of ML D are called (poly-)modal formulas, or briefly, formulas. Formulas of the form p or p are called literals. In case the set D is a singleton, we speak of the language ML of basic modal logic or monomodal logic; in this case we will denote the modal operators by 3 and 2, respectively. Given a finite set P of propositional variables, we let ML D (P) denote the set of formulas in which only variables from P occur. Often the sets P and D are implicitly understood, and suppressed in the notation. Generally it will su ce to treat examples, proofs, etc., from monomodal logic. Remark 1.6 The negation ' of a formula ' can inductively be defined as follows:? := > > :=? p := p p := p (' _ ) := ' ^ (' ^ ) := ' _ 2 d ' := 3 d ' 3 d ' := 2 d ' On the basis of this, we can also define the other standard abbreviated connectives, such as! and $. We assume that the reader is familiar with standard syntactic notions such as those of a subformula or the construction tree of a formula, and with standard syntactic operations such as substitution. Concerning the latter, we let '[ /p] denote the formula that we obtain by substituting all occurrences of p in ' by.

3 Lectures on the modal µ-calculus 5 Definition 1.7 e define the collection Sfor( ) of subformulas of a modal formula ' by the following induction on the complexity of ': Sfor(?) := {?} Sfor(>) := {>} Sfor(p) := {p} Sfor( p) := {p, p} FV ('? ) := {'? }[Sfor(') [ Sfor( ) where? 2 {_, ^} Sfor(~') := {~'}[Sfor(') where ~2{3 d, 2 d d 2 D} We write ' P to denote that ' is a subformula of. Thesize of a formula is defined as the number of its subformulas, := Sfor( ). Semantics The relational semantics of modal logic is well known. The basic idea is that the modal operators 3 d and 2 d are both interpreted using the accessibility relation R d. The notion of truth (or satisfaction) is defined as follows. Definition 1.8 Let S = hs, i be a transition system of type (P, D). Then the satisfaction relation between states of S and formulas of ML D (P) is defined by the following formula induction. S,s p if s 2 V (p), S,s p if s 62 V (p), S,s? never, S,s > always, S,s ' _ if S,s ' or S,s, S,s ' ^ if S,s ' and S,s, S,s 3 d ' if S,t ' for some t 2 R d [s], S,s 2 d ' if S,t ' for all t 2 R d [s]. We say that ' is true or holds at s if S,s ', and we let the set ['] S := {s 2 S S,s '}. denote the meaning or extension of ' in S. Alternatively (but equivalently), one may define the semantics of modal formulas directly in terms of this meaning function ['] S. This approach has some advantages in the context of fixpoint operators, since it brings out the role of the powerset algebra }(S) more clearly.

4 6 Modal logic Remark 1.9 Fix an LTS S, thendefine[[']] S by induction on the complexity of ': [p] S = V (p) [ p]] S = S \ V (p) [?]] S =? [>]] S = S [' _ ]] S = [[']] S [ [[ ]] S [' ^ ]] S = [[']] S \ [[ ]] S [3 d ']] S = hr d i[[']] S [2 d ']] S = [R d ][[']] S Here the operations hr d i and [R d ] on }(S) are defined by putting hri(x) := {s 2 S R d [s] \ X 6=?} [R](X) := {s 2 S R d [s] X}. The satisfaction relation may be recovered from this by putting S,s ' i s 2 [[']] S. Definition 1.10 Let s and s 0 be two states in the transition systems S and S 0 of type (P, D), respectively. Then we say that s and s 0 are modally equivalent, notation: S,s! (P,D) S 0,s 0,if s and s 0 satisfy the same modal formulas, that is, S,s ' i S 0,s 0 ', for all modal formulas ' 2 ML D (P). Flows, trees and streams In some parts of these notes deterministic transition systems feature prominently. Definition 1.11 A transition system S = hs, V, Ri is called deterministic if each R d [s] isa singleton. Note that our definition of determinism does not allow R d =? for any point s. Wefirst consider the monomodal case. Definition 1.12 Let P be a set of proposition letters. A deterministic monomodal Kripke model for this language is called a flow model for P, or a }(P)-flow. In case such a structure is of the form h!, V, Succi, wheresucc is the standard successor relation on the set! of natural numbers, we call the structure a stream model for P, or a }(P)-stream. In case the set D of actions is finite, we may just as well identify it with the set k = {0,...,k 1}, wherek is the size of D. We usually restrict to the binary case, that is, k = 2. Our main interest will be in Kripke models that are based on the binary tree, i.e.,atreein which every node has exactly two successors, a left and a right one. Definition 1.13 With 2 = {0, 1}, welet2 denote the set of finite strings of 0s and 1s. We let denote the empty string, while the left- and right successor of a node s are denoted by s 0 and s 1, respectively. Written as a relation, we put Succ i = {(s, s i) s 2 2 }.

5 Lectures on the modal µ-calculus 7 A binary tree over P, or a binary }(P)-tree is a Kripke model of the form h2,v,succ 0, Succ 1 i. Remark 1.14 In the general case, the k-ary tree is the structure (k, Succ 0,...,Succ k 1 ), where k is the set of finite sequences of natural numbers smaller than k, and Succ i is the i-th successor relation given by Succ i = {(s, s i) s 2 k }. A k-flow model is a Kripke model S = hs, V, Ri with k many deterministic accessibility relations, and a k-ary tree model is a k-flow model which is based on the k-ary tree. In deterministic transition systems, the distinction between boxes and diamonds evaporates. It is then convenient to use a single symbol i to denote either the box or the diamond. Definition 1.15 The set MFL k (P) of formulas of k-ary Modal Flow Logic in P is given as follows: ' ::= p p? > ' _ ' ' ^ ' i' where p 2 P, and i<k. In case k = 1 we will also speak of modal stream logic, notation: MSL(P). 1.2 Game semantics We will now describe the semantics defined above in game-theoretic terms. That is, we will define the evaluation game E(,S) associated with a (fixed) formula and a (fixed) LTS S. This game is an example of a board game. In a nutshell, board games are games in which the players move a token along the edge relation of some graph, so that a match of the game corresponds to a (finite or infinite) path through the graph. Furthermore, the winning conditions of a match are determined by the nature of this path. We will meet many examples of board games in these notes, and in Chapter 5 we will study them in more detail. The evaluation game E(,S) isplayedbytwoplayers: Éloise (9 or 0) and Abélard (8 or 1). Given a player, we always denote the opponent of by. As mentioned, a match of the game consists of the two players moving a token from one position to another. Positions are of the form (', s), with ' a subformula of, and s a state of S. It is useful to assign goals to both players: in an arbitrary position (', s), think of 9 trying to show that ' is true at s in S, and of 8 of trying to convince her that ' is false at s. Depending on the type of the position (more precisely, on the formula part of the position), one of the two players may move the token to a next position. For instance, in a position of the form (3 d ', s), it is 9 s turn to move, and she must choose an arbitrary d-successor t of s, thus making (', t) the next position. Intuitively, the idea is that in order to show that 3' is true at s, 9 has to come up with a successor of s where ' holds. Formally, we say that the set of (admissible) next positions that 9 may choose from is given as the set {(', t) t 2 R d [s]}. In the case there is no successor of s to choose, she immediately loses the game. This is a convenient way to formulate the rules for winning and losing this game: if a position (', s)

6 8 Modal logic Position Player Admissible moves (' 1 _ ' 2,s) 9 {(' 1,s),(' 2,s)} (' 1 ^ ' 2,s) 8 {(' 1,s),(' 2,s)} (3 d ', s) 9 {(', t) t 2 R d [s]} (2 d ', s) 8 {(', t) t 2 R d [s]} (?,s) 9? (>,s) 8? (p, s),s2 V (p) 8? (p, s),s62 V (p) 9? ( p, s),s62 V (p) 8? ( p, s),s2 V (p) 9? Table 1: Evaluation game for modal logic has no admissible next positions, the player whose turn it is to play at (', s) gets stuck and immediately loses the game. This convention gives us a nice handle on positions of the form (p, s) wherep is a proposition letter: we always assign to such a position an empty set of admissible moves, but we make 9 responsible for (p, s) in case p is false at s, and 8 in case p is true at s. Inthisway,9 immediately wins if p is true at s, and 8 if it is otherwise. The rules for the negative literals ( p) and the constants,? and >, follow a similar pattern. The full set of rules of the game is given in Table 1. Observe that all matches of this game are finite, since at each move of the game the active formula is reduced in size. (From the general perspective of board games, this means that we need not worry about winning conditions for matches of infinite length.) We may now summarize the game as follows. Definition 1.16 Given a modal formula and a transition system S, the evaluation game E(,S) is defined as the board game given by Table 1, with the set Sfor( ) S providing the positions of the game; that is, a position is a pair consisting of a subformula of and a point in S. The instantiation of this game with starting point (,s) is denoted as E(,S)@(,s). An instance of an evaluation game is a pair consisting of an evaluation game and a starting position of the game. Such an instance will also be called an initialized game, or sometimes, if no confusion is likely, simply a game. A strategy for a player in an initialized game is a method that uses to select his moves during the play. Such a strategy is winning for if every match of the game (starting at the given position) is won by, provided plays according to this strategy. A position (', s) is winning for if has a winning strategy for the game initialized in that position. (This is independent of whether it is s turn to move at the position.) The set of winning positions in E(,S) for is denoted as Win (E(,S)). The main result concerning these games is that they provide an alternative, but equivalent, semantics for modal logic. I Example to be added

7 Lectures on the modal µ-calculus 9 Theorem 1.17 Let be a modal formula, and let S be an LTS. Then for any state s in S it holds that (,s) 2 Win 9 (E(,S)) () S,s. The proof of this Theorem is left to the reader. 1.3 Bisimulations and bisimilarity One of the most fundamental notions in the model theory of modal logic is that of a bisimulation between two transition systems. I discuss bisimilarity as a notion of behavioral equivalence Definition 1.18 Let S and S 0 be two transition systems of the same type (P, D). Then a relation Z S S 0 is a bisimulation of type (P, D) if the following hold, for every (s, s 0 ) 2 Z. (prop) s 2 V (p) i s 0 2 V 0 (p), for all p 2 P; (forth) for all actions d, and for all t 2 R d [s] thereisat 0 2 R 0 d [s0 ]with(t, t 0 ) 2 Z; (back) for all actions d, and for all t 0 2 R 0 d [s0 ]thereisat 2 R d [s] with(t, t 0 ) 2 Z. Two states s and s 0 are called bisimilar, notation: S,s $ P,D S 0,s 0 if there is some bisimulation Z of type (P, D) with(s, s 0 ) 2 Z. If no confusion is likely to arise, we generally drop the subscripts, writing $ rather than $ P,D. Bisimilarity and modal equivalence In order to understand the importance of this notion for modal logic, the starting point should be the observation that the truth of modal formulas is invariant under bisimilarity. Recall that! denotes the relation of modal equivalence. Theorem 1.19 (Bisimulation Invariance) Let S and S 0 be two transition systems of the same type. Then S,s $ S 0,s 0 ) S,s! S 0,s 0 for every pair of states s in S and s 0 in S 0. Proof. By a straightforward induction on the complexity of modal formulas one proves that bisimilar states satisfy the same formulas. qed But there is much more to say about the relation between modal logic and bisimilarity than Theorem In particular, for some classes of models, one may prove a converse statement, which amounts to saying that the notions of bisimilarity and modal equivalence coincide. Such classes are said to have the Hennessy-Milner property. As an example we mention the class of finitely branching transition systems.

8 10 Modal logic Theorem 1.20 (Hennessy-Milner Property) Let S and S 0 be two finitely branching transition systems of the same type. Then for every pair of states s in S and s 0 in S 0. S,s $ S 0,s 0 () S,s! S 0,s 0 Proof. The direction from left to right follows from Theorem In order to prove the opposite direction, one may show that the relation! of modal equivalence itself is a bisimulation. Details are left to the reader. qed This theorem can be read as indication of the expressiveness of modal logic: any di erence in behaviour between two states in finitely branching transition systems can in fact be witnessed by a concrete modal formula. As another witness to this expressivity, in section 1.5 we will see that modal logic is su ciently rich to express all bisimulation-invariant first-order properties. Obviously, this result also adds considerable strength to the link between modal logic and bisimilarity. As a corollary of the bisimulation invariance theorem, modal logic has the tree model property, that is, every satisfiable modal formula is satisfiable on a structure that has the shape of a tree. Definition 1.21 A transition system S of type (P, D) is called tree-like if the structure hs, S d2d R di is a tree. The key step in the proof of the tree model property of modal logic is the observation that every transition system can be unravelled into a bisimilar tree-like model. The basic idea of such an unravelling is the new states encode (part of) the history of the old states. Technically, the new states are the paths through the old system. Definition 1.22 Let S = hs, V, Ri be a transition system of type (P, D). A (finite) path through S is a nonempty sequence of the form (s 0,d 1,s 1,...,d n,s n ) such that R di s i 1 s i for all i with 0 <iapplen. The set of paths through S is denoted as Paths (S); we use the notation Paths s (S) for the set of paths starting at s. The unravelling of S around a state s is the transition system ~ S s which is coalgebraically defined as the structure hpaths s (S),~i, where the coalgebra map ~ =(~ V, (~ d d 2 D)) is given by putting ~ V (s 0,d 1,s 1,...,d n,s n ) := V (s n ), ~ d (s 0,d 1,s 1,...,d n,s n ) := {(s 0,d 1,s 1,...,d n,s n,d,t) 2 Paths s (S) R d s n t}. Finally, the unravelling of a pointed transition system (S,s) is the pointed structure ( ~ S s,s), where (with some abuse of notation) we let s denote the path of length zero that starts and finishes at s. Clearly, unravellings are tree-like structures, and any pointed transition system is bisimilar to its unravelling. But then the following theorem is immediate by Theorem Theorem 1.23 (Tree Model Property) Let ' be a satisfiable modal formula. Then ' is satisfiable at the root of a tree-like model.

9 Lectures on the modal µ-calculus 11 Bisimilarity game We may also give a game-theoretic characterization of the notion of bisimilarity. We first give an informal description of the game that we will employ. A match of the bisimilarity game between two Kripke models S and S 0 is played by two players, 9 and 8. As in the evaluation game, these players move a token around from one position of the game to the next one. In the game there are two kinds of positions: pairs of the form (s, s 0 ) 2 S S 0 are called basic positions and belong to 9. The other positions are of the form Z S S 0 and belong to 8. The idea of the game is that at a position (s, s 0 ), 9 claims that s and s 0 are bisimilar, and to substantiate this claim she proposes a local bisimulation Z S S 0 (see below) for s and s 0. This relation Z can be seen as providing a set of witnesses for 9 s claim that s and s 0 are bisimilar. Implicitly, 9 s claim at a position Z S S 0 is that all pairs in Z are bisimilar, so 8 can pick an arbitrary pair (t, t 0 ) 2 Z and challenge 9 to show that these t and t 0 are bisimilar. Definition 1.24 Let S and S 0 be two transition systems of the same type (P, D). Then a relation Z S S 0 is a local bisimulation for two points s 2 S and s 0 2 S 0, if it satisfies the properties (prop), (back) and (forth) of Definition 1.18 for this specific s and s 0. If a player gets stuck in a match of this game, then the opponent wins the match. For instance, if s and s 0 disagree about some proposition letter, then there is no local bisimulation for s and s 0, and so the corresponding position (s, s 0 ) is an immediate loss for 9. Or,ifneither s nor s 0 has successors, and agree on the truth of all proposition letters, then 9 could choose the empty relation as a local bisimulation, so that 8 would lose the match at his next move. A new option arises if neither player gets stuck: this game may also have matches that last forever. Nevertheless, we can still declare a winner for such matches, and the agreement is that 9 is the winner of any infinite match. Formally, we put the following. Definition 1.25 The bisimilarity game B(S, S 0 )betweentwokripkemodelss and S 0 is the board game given by Table 2, with the winning condition that finite matches are lost by the player who got stuck, while all infinite matches are won by 9. A position (s, s 0 )iswinning for if has a winning strategy for the game initialized in that position. The set of these positions is denoted as Win (B(S, S 0 )). Position Player Admissible moves (s, s 0 ) 2 S S 0 9 {Z 2 }(S S 0 ) Z is a local bisimulation for s and s 0 } Z 2 }(S S 0 ) 8 Z = {(t, t 0 ) (t, t 0 ) 2 Z} Table 2: Bisimilarity game for Kripke models Also observe that a bisimulation is a relation which is a local bisimulation for each of its members. The following theorem states that the collection of basic winning positions for 9 forms the largest bisimulation between S and S 0.

10 12 Modal logic Theorem 1.26 Let (S,s) and (S 0,s 0 ) be two pointed Kripke models. Then S,s $ S 0,s 0 i (s, s 0 ) 2 Win 9 (B(S, S 0 )). Proof. For the direction from left to right: suppose that Z is a bisimulation between S and S 0 linking s and s 0. Suppose that 9, starting from position (s, s 0 ), always chooses the relation Z itself as the local bisimulation. A straightforward verification, by induction on the length of the match, shows that this strategy always provides her with a legitimate move, and that it keeps her alive forever. This proves that it is a winning strategy. For the converse direction, it su ces to show that the relation {(t, t 0 ) 2 S S 0 (t, t 0 ) 2 Win 9 (B(S, S 0 ))} itself is in fact a bisimulation. We leave the details for the reader. qed Remark 1.27 game. I The bisimilarity game should not be confused with the bisimulation Bisimulations via relation lifting Together, the back- and forth clause of the definition of a bisimulation express that the pair of respective successor sets of two bisimilar states must belong to the so-called Egli-Milner lifting }Z of the bisimulation Z. In fact, the notion of a bisimilation can be completely defined in terms of relation lifting. Definition 1.28 Given a relation Z A A 0, define the relation }Z }A }A 0 as follows: }Z := {(X, X 0 ) for all x 2 X there is an x 0 2 X 0 with (x, x 0 ) 2 Z & for all x 0 2 X 0 there is an x 2 X with (x, x 0 ) 2 Z}. Similarly, define, for a Kripke functor K = K D,P, the relation KZ KA KA 0 as follows: KZ := {((, X), ( 0,X 0 )) = 0 and (X d,xd 0 ) 2 }Z for each d 2 D}. The relations }Z and KZ are called the lifting of Z with respect to } and K, respectively. We say that Z A A 0 is full on B 2 }A and B 0 2 }A 0 if (B,B 0 ) 2 }Z. It is completely straightforward to check that a nonempty relation Z linking two transition systems S and S 0 is a local bisimulation for two states s and s 0 i ( (s), 0 (s 0 )) 2 KZ. In particular, 9 s move in the bisimilarity game at a position (s, s 0 ) consists of choosing a binary relation Z such that ( (s), 0 (s 0 )) 2 KZ. The following characterization of bisimulations is also an immediate consequence. Proposition 1.29 Let S and S 0 be two Kripke coalgebras for some Kripke functor K, and let Z S S 0 be some relation. Then Z is a bisimulation i ( (s), 0 (s 0 )) 2 KZ for all (s, s 0 ) 2 Z. (1)

11 Lectures on the modal µ-calculus Finite models and computational aspects I complexity of model checking I filtration & polysize model property I complexity of satisfiability I complexity of global consequence 1.5 Modal logic and first-order logic I modal logic is the bisimulation invariant fragment of first-order logic 1.6 Completeness 1.7 The cover modality As we will see now, there is an interesting alternative for the standard formulation of basic modal logic in terms of boxes and diamonds. This alternative set-up is based on a connective which turns sets of formulas into formulas. Definition 1.30 Let be a finite set of formulas. Then r is a formula, which holds at a state s in a Kripke model if every formula in holds at some successor of s, while at the same time, every successor of s makes some formula in true. The operator r is called the cover modality. It is not so hard to see that the cover modality can be defined in the standard modal language: r 2 _ ^ ^ 3, (2) where 3 denotes the set {3' ' 2 }. Things start to get interesting once we realize that both the ordinary diamond 3 and the ordinary box 2 can be expressed in terms of the cover modality (and the disjunction): 3' r{', >}, 2' r? _r{'}. (3) Here, as always, we use the convention that W? =? and V? = >. Remark 1.31 Observe that this definition involves the 89&89 pattern that we know from the notion of relation lifting } defined in the previous section. In other words, the semantics of the cover modality can be expressed in terms of relation lifting. For that purpose, observe that we may think of the forcing or satisfaction relation simply as a binary relation between states and formulas. The we find that S,s r i ( R (s), ) 2 }( ). for any pointed Kripke model (S,s) and any finite set of formulas.

12 14 Modal logic Given that r and {3, 2} are mutually expressible, we arrive at the following definition and proposition. Definition 1.32 Formulas of the language ML r are given by the following recursive definition: ' ::= p p? > ' _ ' ' ^ ' r where denotes a finite set of formulas. Proposition 1.33 The languages ML and ML r are equally expressive. Proof. Immediate by (2) and (3). qed The real importance of the cover modality is that it allows us to almost completely eliminate the Boolean conjunction. This remarkable fact is based on the following modal distributive law. Recall from Definition 1.28 that a relation Z A A 0 is full on A and A 0 if (A, A 0 ) 2 }Z. Proposition 1.34 Let and 0 be two sets of formulas. Then the following two formulas are equivalent: r ^r 0 _ {r Z Z is full on and 0 }, (4) where, given a relation Z 0, we define Z := {' ^ ' 0 (', ' 0 ) 2 Z}. Proof. For the direction from left to right, suppose that S,s r ^r 0. Let Z 0 consist of those pairs (', ' 0 ) such that the conjunction ' ^ ' 0 is true at some successor t of s. It is then straightforward to verify that Z is full on and 0, and that S,s r Z. The converse direction follows fairly directly from the definitions. qed As a corollary of Proposition 1.34 we can restrict the use of conjunction in modal logic to that of a special conjunction connective which may only be applied to a propositional formula and a certain conjunction of r-formula. Definition 1.35 Fix finite sets P of proposition letters and D of atomic actions, respectively. We first define the set CL(P) of literal conjunctions by the following grammar: ::= p p? > ^. Next, let = { d d 2 D} be a D-indexed family of formulas, and write r D := V d2d r d d, where r d is the cover modality associated with the accessibility relation R d of d. Finally, the following grammar: ' ::=? > ' _ ' r D. defines the set DML D (P) of disjunctive polymodal formulas in D and P.

13 Lectures on the modal µ-calculus 15 The following theorem states that every modal formula can be rewritten into an equivalent disjunctive normal form. Theorem 1.36 For any P and D, the languages ML D (P) and DML D (P) are expressively equivalent. We leave the proof of this result as an exercise to the reader. Notes Modal logic has a long history in philosophy and mathematics, for an overview we refer to Blackburn, de Rijke and Venema [3]. The use of modal formalisms as specification languages in process theory goes back at least to the 1970s, with Pratt [26] and Pnueli [25] being two influential early papers. The notion of bisimulation, which plays an important role in modal logic and process theory alike, was first introduced in a modal logic context by van Benthem [2], who proved that modal logic is the bisimulation invariant fragment of first-order logic. The notion was later, but independently, introduced in a process theory setting by Park [24]. At the time of writing we do not know who first took a game-theoretical perspective on the semantics of modal logic. The cover modality r was introduced independently by Moss [17] and Janin & Walukiewicz [10]. Readers who want to study modal logic in more detail are referred to Blackburn, de Rijke and Venema [3] or Chagrov & Zakharyaschev [5]. Exercises Exercise 1.1 Prove Theorem Exercise 1.2 (bisimilarity game) Consider the following version B! (S, S 0 ) of the bisimilarity game between two transition systems S and S 0. Positions of this game are of the form either (s, s 0, 8, ), (s, s 0, 9, ) or (Z, ), with s 2 S, s 0 2 S 0, Z S S 0 and either a natural number or!. The admissible moves for 9 and 8 are displayed in the following table: Position Player Admissible moves (s, s 0, 8, ) 8 {(s, s 0, 9, ) < } (s, s 0, 9, ) 9 {(Z, ) Z is a local bisimulation for s and s 0 } (Z, ) 8 {(s, s 0, 8, ) (s, s 0 ) 2 Z} Note that all matches of this game have finite length. We write S,s $ S 0,s 0 to denote that 9 has a winning strategy in the game B! (S, S 0 ) starting at position (s, s 0, 8, ). It is not hard to see that S,s $! S 0,s 0 i S,s $ k S 0,s 0 for all k<!. (a) Give concrete examples such that S,s $! S 0,s 0 but not S,s $ S 0,s 0. (Hint: think of two modally equivalent but not bisimilar states.)

14 16 Modal logic (b) Let k 0 be a natural number. Prove that, for all S,s and S 0,s 0 : S,s $ k S 0,s 0 ) S,s! k S 0,s 0. Here! k denotes the modal equivalence relation with respect to formulas of modal depth at most k. Here we use a slightly nonstandard notion of modal depth, defined as follows: d(?),d(>) := 0, d(p) := 1 for p 2 P, d(' ^ ),d(' _ ) := max(d('),d( )), and d(3'),d(2') :=1+d('). (c) Let S and S 0 be finitely branching transition systems. Prove directly (i.e., without using part (b)) that (i) ) (ii), for all s 2 S and s 0 2 S 0 : (i) S,s $! S 0,s 0 (ii) S,s $ S 0,s 0. (d) Does the implication in (c) hold in the case that only one of the two transition systems is finitely branching? Exercise 1.3 Let and be finite sets of formulas. Prove that Exercise 1.4 Prove Theorem r [{ W } _ r [ 0? 6= 0.

CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

More information

Mathematical Induction

Mathematical Induction Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How

More information

Which Semantics for Neighbourhood Semantics?

Which Semantics for Neighbourhood Semantics? Which Semantics for Neighbourhood Semantics? Carlos Areces INRIA Nancy, Grand Est, France Diego Figueira INRIA, LSV, ENS Cachan, France Abstract In this article we discuss two alternative proposals for

More information

Regular Languages and Finite Automata

Regular Languages and Finite Automata Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a

More information

Testing LTL Formula Translation into Büchi Automata

Testing LTL Formula Translation into Büchi Automata Testing LTL Formula Translation into Büchi Automata Heikki Tauriainen and Keijo Heljanko Helsinki University of Technology, Laboratory for Theoretical Computer Science, P. O. Box 5400, FIN-02015 HUT, Finland

More information

Full and Complete Binary Trees

Full and Complete Binary Trees Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

More information

A Propositional Dynamic Logic for CCS Programs

A Propositional Dynamic Logic for CCS Programs A Propositional Dynamic Logic for CCS Programs Mario R. F. Benevides and L. Menasché Schechter {mario,luis}@cos.ufrj.br Abstract This work presents a Propositional Dynamic Logic in which the programs are

More information

Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

6.3 Conditional Probability and Independence

6.3 Conditional Probability and Independence 222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

More information

Reading 13 : Finite State Automata and Regular Expressions

Reading 13 : Finite State Automata and Regular Expressions CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

More information

CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1. Peter Kahn. Spring 2007 CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

More information

Lecture 17 : Equivalence and Order Relations DRAFT

Lecture 17 : Equivalence and Order Relations DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

More information

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.

This asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements. 3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but

More information

WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS. Christopher Heil Georgia Institute of Technology WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015

ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015 ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1

More information

Non-deterministic Semantics and the Undecidability of Boolean BI

Non-deterministic Semantics and the Undecidability of Boolean BI 1 Non-deterministic Semantics and the Undecidability of Boolean BI DOMINIQUE LARCHEY-WENDLING, LORIA CNRS, Nancy, France DIDIER GALMICHE, LORIA Université Henri Poincaré, Nancy, France We solve the open

More information

So let us begin our quest to find the holy grail of real analysis.

So let us begin our quest to find the holy grail of real analysis. 1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

More information

Basic Proof Techniques

Basic Proof Techniques Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

More information

Cartesian Products and Relations

Cartesian Products and Relations Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

More information

Model Checking: An Introduction

Model Checking: An Introduction Announcements Model Checking: An Introduction Meeting 2 Office hours M 1:30pm-2:30pm W 5:30pm-6:30pm (after class) and by appointment ECOT 621 Moodle problems? Fundamentals of Programming Languages CSCI

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

More information

C H A P T E R Regular Expressions regular expression

C H A P T E R Regular Expressions regular expression 7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun

More information

(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.

(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine. (LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of first-order logic will use the following symbols: variables connectives (,,,,

More information

Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence

More information

Mathematical Induction

Mathematical Induction Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

More information

Lecture 16 : Relations and Functions DRAFT

Lecture 16 : Relations and Functions DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

More information

Correspondence analysis for strong three-valued logic

Correspondence analysis for strong three-valued logic Correspondence analysis for strong three-valued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K 3 ).

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

More information

CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

More information

Goal of the Talk Theorem The class of languages recognisable by T -coalgebra automata is closed under taking complements.

Goal of the Talk Theorem The class of languages recognisable by T -coalgebra automata is closed under taking complements. Complementation of Coalgebra Automata Christian Kissig (University of Leicester) joint work with Yde Venema (Universiteit van Amsterdam) 07 Sept 2009 / Universitá degli Studi di Udine / CALCO 2009 Goal

More information

Logic in Computer Science: Logic Gates

Logic in Computer Science: Logic Gates Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers

More information

3 Some Integer Functions

3 Some Integer Functions 3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

More information

University of Ostrava. Reasoning in Description Logic with Semantic Tableau Binary Trees

University of Ostrava. Reasoning in Description Logic with Semantic Tableau Binary Trees University of Ostrava Institute for Research and Applications of Fuzzy Modeling Reasoning in Description Logic with Semantic Tableau Binary Trees Alena Lukasová Research report No. 63 2005 Submitted/to

More information

NP-Completeness and Cook s Theorem

NP-Completeness and Cook s Theorem NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:

More information

8 Divisibility and prime numbers

8 Divisibility and prime numbers 8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Likewise, we have contradictions: formulas that can only be false, e.g. (p p).

Likewise, we have contradictions: formulas that can only be false, e.g. (p p). CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula

More information

Predicate Logic. For example, consider the following argument:

Predicate Logic. For example, consider the following argument: Predicate Logic The analysis of compound statements covers key aspects of human reasoning but does not capture many important, and common, instances of reasoning that are also logically valid. For example,

More information

ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS

ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2012 1 p. 43 48 ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and Information

More information

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

More information

One pile, two pile, three piles

One pile, two pile, three piles CHAPTER 4 One pile, two pile, three piles 1. One pile Rules: One pile is a two-player game. Place a small handful of stones in the middle. At every turn, the player decided whether to take one, two, or

More information

Automata on Infinite Words and Trees

Automata on Infinite Words and Trees Automata on Infinite Words and Trees Course notes for the course Automata on Infinite Words and Trees given by Dr. Meghyn Bienvenu at Universität Bremen in the 2009-2010 winter semester Last modified:

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Decision Making under Uncertainty

Decision Making under Uncertainty 6.825 Techniques in Artificial Intelligence Decision Making under Uncertainty How to make one decision in the face of uncertainty Lecture 19 1 In the next two lectures, we ll look at the question of how

More information

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

More information

CMPSCI 250: Introduction to Computation. Lecture #19: Regular Expressions and Their Languages David Mix Barrington 11 April 2013

CMPSCI 250: Introduction to Computation. Lecture #19: Regular Expressions and Their Languages David Mix Barrington 11 April 2013 CMPSCI 250: Introduction to Computation Lecture #19: Regular Expressions and Their Languages David Mix Barrington 11 April 2013 Regular Expressions and Their Languages Alphabets, Strings and Languages

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

The theory of the six stages of learning with integers (Published in Mathematics in Schools, Volume 29, Number 2, March 2000) Stage 1

The theory of the six stages of learning with integers (Published in Mathematics in Schools, Volume 29, Number 2, March 2000) Stage 1 The theory of the six stages of learning with integers (Published in Mathematics in Schools, Volume 29, Number 2, March 2000) Stage 1 Free interaction In the case of the study of integers, this first stage

More information

o-minimality and Uniformity in n 1 Graphs

o-minimality and Uniformity in n 1 Graphs o-minimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER

THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER JASPER DEANTONIO Abstract. This paper is a study of the Turing Degrees, which are levels of incomputability naturally arising from sets of natural numbers.

More information

Regular Expressions and Automata using Haskell

Regular Expressions and Automata using Haskell Regular Expressions and Automata using Haskell Simon Thompson Computing Laboratory University of Kent at Canterbury January 2000 Contents 1 Introduction 2 2 Regular Expressions 2 3 Matching regular expressions

More information

Lecture 1. Basic Concepts of Set Theory, Functions and Relations

Lecture 1. Basic Concepts of Set Theory, Functions and Relations September 7, 2005 p. 1 Lecture 1. Basic Concepts of Set Theory, Functions and Relations 0. Preliminaries...1 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2

More information

Math 3000 Section 003 Intro to Abstract Math Homework 2

Math 3000 Section 003 Intro to Abstract Math Homework 2 Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these

More information

Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy

Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy Kim S. Larsen Odense University Abstract For many years, regular expressions with back referencing have been used in a variety

More information

1 Operational Semantics for While

1 Operational Semantics for While Models of Computation, 2010 1 1 Operational Semantics for While The language While of simple while programs has a grammar consisting of three syntactic categories: numeric expressions, which represent

More information

Model Checking II Temporal Logic Model Checking

Model Checking II Temporal Logic Model Checking 1/32 Model Checking II Temporal Logic Model Checking Edmund M Clarke, Jr School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 2/32 Temporal Logic Model Checking Specification Language:

More information

[Refer Slide Time: 05:10]

[Refer Slide Time: 05:10] Principles of Programming Languages Prof: S. Arun Kumar Department of Computer Science and Engineering Indian Institute of Technology Delhi Lecture no 7 Lecture Title: Syntactic Classes Welcome to lecture

More information

2. The Language of First-order Logic

2. The Language of First-order Logic 2. The Language of First-order Logic KR & R Brachman & Levesque 2005 17 Declarative language Before building system before there can be learning, reasoning, planning, explanation... need to be able to

More information

MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. III - Logic and Computer Science - Phokion G. Kolaitis

MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. III - Logic and Computer Science - Phokion G. Kolaitis LOGIC AND COMPUTER SCIENCE Phokion G. Kolaitis Computer Science Department, University of California, Santa Cruz, CA 95064, USA Keywords: algorithm, Armstrong s axioms, complete problem, complexity class,

More information

Automata-based Verification - I

Automata-based Verification - I CS3172: Advanced Algorithms Automata-based Verification - I Howard Barringer Room KB2.20: email: howard.barringer@manchester.ac.uk March 2006 Supporting and Background Material Copies of key slides (already

More information

A first step towards modeling semistructured data in hybrid multimodal logic

A first step towards modeling semistructured data in hybrid multimodal logic A first step towards modeling semistructured data in hybrid multimodal logic Nicole Bidoit * Serenella Cerrito ** Virginie Thion * * LRI UMR CNRS 8623, Université Paris 11, Centre d Orsay. ** LaMI UMR

More information

A Little Set Theory (Never Hurt Anybody)

A Little Set Theory (Never Hurt Anybody) A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra

More information

Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.

Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as true

More information

Software Modeling and Verification

Software Modeling and Verification Software Modeling and Verification Alessandro Aldini DiSBeF - Sezione STI University of Urbino Carlo Bo Italy 3-4 February 2015 Algorithmic verification Correctness problem Is the software/hardware system

More information

Graph Theory Problems and Solutions

Graph Theory Problems and Solutions raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

More information

Mathematical Induction. Lecture 10-11

Mathematical Induction. Lecture 10-11 Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

More information

UPDATES OF LOGIC PROGRAMS

UPDATES OF LOGIC PROGRAMS Computing and Informatics, Vol. 20, 2001,????, V 2006-Nov-6 UPDATES OF LOGIC PROGRAMS Ján Šefránek Department of Applied Informatics, Faculty of Mathematics, Physics and Informatics, Comenius University,

More information

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part

More information

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

More information

http://aejm.ca Journal of Mathematics http://rema.ca Volume 1, Number 1, Summer 2006 pp. 69 86

http://aejm.ca Journal of Mathematics http://rema.ca Volume 1, Number 1, Summer 2006 pp. 69 86 Atlantic Electronic http://aejm.ca Journal of Mathematics http://rema.ca Volume 1, Number 1, Summer 2006 pp. 69 86 AUTOMATED RECOGNITION OF STUTTER INVARIANCE OF LTL FORMULAS Jeffrey Dallien 1 and Wendy

More information

CS510 Software Engineering

CS510 Software Engineering CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15-cs510-se

More information

Near Optimal Solutions

Near Optimal Solutions Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

More information

On strong fairness in UNITY

On strong fairness in UNITY On strong fairness in UNITY H.P.Gumm, D.Zhukov Fachbereich Mathematik und Informatik Philipps Universität Marburg {gumm,shukov}@mathematik.uni-marburg.de Abstract. In [6] Tsay and Bagrodia present a correct

More information

Row Echelon Form and Reduced Row Echelon Form

Row Echelon Form and Reduced Row Echelon Form These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005

Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005 Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable

More information

GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

More information

Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic

Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic Lars Birkedal Rasmus Ejlers Møgelberg Rasmus Lerchedahl Petersen IT University Technical Report Series TR-00-7 ISSN 600 600 February 00

More information

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

More information

A Hennessy-Milner Property for Many-Valued Modal Logics

A Hennessy-Milner Property for Many-Valued Modal Logics A Hennessy-Milner Property for Many-Valued Modal Logics Michel Marti 1 Institute of Computer Science and Applied Mathematics, University of Bern Neubrückstrasse 10, CH-3012 Bern, Switzerland mmarti@iam.unibe.ch

More information

The Classes P and NP

The Classes P and NP The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the

More information

LEARNING OBJECTIVES FOR THIS CHAPTER

LEARNING OBJECTIVES FOR THIS CHAPTER CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

More information

Graphing calculators Transparencies (optional)

Graphing calculators Transparencies (optional) What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used

More information

How To Understand The Theory Of Computer Science

How To Understand The Theory Of Computer Science Theory of Computation Lecture Notes Abhijat Vichare August 2005 Contents 1 Introduction 2 What is Computation? 3 The λ Calculus 3.1 Conversions: 3.2 The calculus in use 3.3 Few Important Theorems 3.4 Worked

More information

MetaGame: An Animation Tool for Model-Checking Games

MetaGame: An Animation Tool for Model-Checking Games MetaGame: An Animation Tool for Model-Checking Games Markus Müller-Olm 1 and Haiseung Yoo 2 1 FernUniversität in Hagen, Fachbereich Informatik, LG PI 5 Universitätsstr. 1, 58097 Hagen, Germany mmo@ls5.informatik.uni-dortmund.de

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

COMPUTER SCIENCE TRIPOS

COMPUTER SCIENCE TRIPOS CST.98.5.1 COMPUTER SCIENCE TRIPOS Part IB Wednesday 3 June 1998 1.30 to 4.30 Paper 5 Answer five questions. No more than two questions from any one section are to be answered. Submit the answers in five

More information

3515ICT Theory of Computation Turing Machines

3515ICT Theory of Computation Turing Machines Griffith University 3515ICT Theory of Computation Turing Machines (Based loosely on slides by Harald Søndergaard of The University of Melbourne) 9-0 Overview Turing machines: a general model of computation

More information

The Halting Problem is Undecidable

The Halting Problem is Undecidable 185 Corollary G = { M, w w L(M) } is not Turing-recognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine

More information

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010 CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

More information

Handout #1: Mathematical Reasoning

Handout #1: Mathematical Reasoning Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

More information

Solutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1

Solutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1 Solutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1 Prof S Bringsjord 0317161200NY Contents I Problems 1 II Solutions 3 Solution to Q1 3 Solutions to Q3 4 Solutions to Q4.(a) (i) 4 Solution to Q4.(a)........................................

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information